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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 13 — Jul. 1, 2013
  • pp: 15409–15417
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Estimating modal instability threshold for photonic crystal rod fiber amplifiers

Mette Marie Johansen, Kristian Rymann Hansen, Marko Laurila, Thomas Tanggaard Alkeskjold, and Jesper Lægsgaard  »View Author Affiliations


Optics Express, Vol. 21, Issue 13, pp. 15409-15417 (2013)
http://dx.doi.org/10.1364/OE.21.015409


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Abstract

We present a semi-analytic numerical model to estimate the transverse modal instability (TMI) threshold for photonic crystal rod amplifiers. The model includes thermally induced waveguide perturbations in the fiber cross section modeled with finite element simulations, and the relative intensity noise (RIN) of the seed laser, which seeds mode coupling between the fundamental and higher order mode. The TMI threshold is predicted to ~370 W – 440 W depending on RIN for the distributed modal filtering rod fiber.

© 2013 OSA

1. Introduction

2. Thermally induced waveguide perturbation and mode distributions

Operating temperatures of fiber amplifiers increase with increasing extracted output power per unit length. The heat originates from the energy difference between pump and signal photons i.e. the quantum defect, and dissipates from the active core towards the fiber boundary, yielding a temperature gradient over the fiber cross section. Rod fiber amplifiers are double clad structures having an outer air cladding guiding high NA pump light. The temperature gradient within the air cladding affects all allowed modes, and perturbs the waveguiding properties especially for core guided modes. The guided mode properties and the thermo-optic coupling between different guided modes are mainly determined by the temperature profile in the core region, which is not significantly affected by the presence of an air cladding. Temperatures on the order of 100°C – 200°C are reached within the air cladding causing significant increments in the refractive indices for large core low NA fibers.

For the purpose of calculating temperature profiles, the PCF structure of the rod fiber is approximated by a set of four concentric cylindrical layers [10

10. E. Coscelli, F. Poli, T. T. Alkeskjold, M. M. Jørgensen, L. Leick, J. Broeng, A. Cucinotta, and S. Selleri, “Thermal Effects on the Single-Mode Regime of Distributed Modal Filtering Rod Fiber,” J. Lightwave Technol. 30(22), 3494–3499 (2012). [CrossRef]

], corresponding to the core,inner cladding, air cladding, and outer fiber, see Fig. 1
Fig. 1 The rod fiber cross section (right) is approximated by four concentric circles representing core, inner cladding, air cladding, and outer fiber (left). The heat load Q0 is assumed uniform across the core.
. The radius of the core is given by the area of the gain medium in the rod fiber amplifier.

The heat gradient is much larger in the radial cross section than along the fiber length, thus for an isotropic medium the steady state heat equation is [11

11. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” J. Quant. Electron. 37(2), 207–217 (2001). [CrossRef]

]
1rr(rT(r)r)=qki,
(1)
where the heat load density q is the heat load Q divided by the area of the active core Acore. Q is Q0 in the core and 0 outside the core, and ki is the thermal conductivity of the i’th layer. The parameters used in the calculations are given in Table 1

Table 1. Parameters Used in This Work

table-icon
View This Table
.

The heat load generated in the gain medium depends on the quantum defect and increases with pump, Ppump [12

12. M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power cw fiber lasers,” Proc. SPIE 7195, 71951U, 71951U-11 (2009). [CrossRef]

]. The slope efficiency S of the rod fiber is included as a reasonable approximation for determining the heat load as a function of signal power, Psignal.
Q0=110αdL/10dL(1λpλs)Ppump110αdL/10dL(1λpλs)PsignalS,
(2)
where α is the pump absorption, λp and λs is the pump and signal wavelength. The fractiondescribes the change in pump power within dL due to pump photons converted to signal photons, and the bracket is the quantum defect for the pump photons. The largest heat load is assumed at the fiber output end in a backwards pumped configuration, where the rod fiber attains the highest stimulated emission and thereby largest quantum defect heating. Hence the last 10 cm of the rod fiber dL is considered for estimating the generated heat load as a function of signal power. The slope efficiency has been measured to 71% – 75% for the DMF85, and is set to 70% in the calculations. The thermal load, and therefore the threshold, for onset of TMI, depends on pump absorption. The small signal pump absorption is typically 20 dB/m for a rod fiber amplifier, but it decreases with increased population of the lasing level. The pump absorption is set to the realistic value of 10 dB/m in the calculations.

Heat diffusion is often compared to electrical charge diffusion and can be thought of as heat flow through a thermal resistance. The solution in one layer of the fiber depends on the boundary condition of the surrounding layer. Inside the core the solution to Eq. (1) is parabolic, when assuming a uniform heat load over the gain medium
T(r)=Tcore+Q04Acoreki(rcore2r2),
(3)
Tcore is the temperature at the core edge, and rcore is the core radius. The solution decays logarithmic outside the active material, where Q = 0 in Eq. (1).
T(r)=Ti+Q02πkiln(rir)
(4)
i represents the current layer, thus Ti is the temperature at the outer boundary, and ri is the outer radius of layer i. The air cladding is thin silica bridges separating air holes, and is approximated by an effective thermal conductivity given by the number of air holes, the air clad width and the width of the silica bridges separating the air holes [13

13. J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express 11(22), 2982–2990 (2003). [CrossRef] [PubMed]

]. The fibers are assumed to be placed in water at temperature T0 with forced convective cooling with coefficient h, yielding a temperature at the fiber edge of
Tfiberedge=T0+Q02πrfiberh
(5)
The temperature increase of the fiber causes a refractive index increase across the entire fiber cross section due to the thermo-optic effect with coefficient η.

Δε=η(T(r)T0)
(6)

Figure 5
Fig. 5 Calculated mode distributions of the FM (top) and first HOM (bottom) for the DMF85 rod fiber for selected signal powers. Larger thermal load causes larger core confinement.
shows the calculated mode distributions of the FM and first HOM for increasing signal powers. The thermal load increases the core refractive index and slowly blueshifts the SM window of the DMF85 for increasing signal power [16

16. M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express 20(5), 5742–5753 (2012). [CrossRef] [PubMed]

]. At some core temperature the MM regime has shifted to the signal wavelength, and the first HOM becomes guided with increasing core overlap, eventually causing TMI.

3. Estimating modal instability threshold

The semi-analytic model by Hansen et al. [8

8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37(12), 2382–2384 (2012). [CrossRef] [PubMed]

,9

9. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21(2), 1944–1971 (2013). [CrossRef] [PubMed]

], describing TMI in fiber amplifiers is combined with FEM calculations to estimate the TMI threshold for PCF fibers. The mode power transfer is determined by a nonlinear coupling constant χ that depends on the frequency difference Ω between the FM and HOM, the bulk gain coefficient g, and the transverse core overlap ratio Г of the FM and HOM determined from the mode calculations. The nonlinear coupling constant χ is given by [8

8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37(12), 2382–2384 (2012). [CrossRef] [PubMed]

,9

9. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21(2), 1944–1971 (2013). [CrossRef] [PubMed]

]
χ(Ω)=ηω2kfiberβc2Im[A(Ω)](1λsλp)
(7)
where η is the thermo-optic coefficient and β is the propagation constant. A is an effective overlap integral that is determined from the mode distributions ψ for the FM and HOM
A(Ω)=ψFM(r)ψHOM(r)coreG(r,r,Ω)ψFM(r)ψHOM(r)d2rd2r
(8)
The inner integral is over the doped core region, and the outer integral is over the full cross section for the transverse coordinates r and r. The Green’s function G is the solution to the Fourier transformed transient heat equation that describes the change in temperature due to the thermal load Q. Assuming G to be translation invariant it is given by
G(rr,Ω)=12πK0[iρCkΩ(rr)],
(9)
where K0 is the zeroth order modified Bessel function of second kind, ρ is the density, and C is the specific heat capacity. G is a convolution operator that allows the inner integral in Eq. (8) to be determined by the convolution theorem. χ in Eq. (7) is an odd function that tends towards zero as Ω approaches zero. Therefore, there must be a seed to initialize power transfer from the FM to the HOM. We consider TMI seeded by relative intensity noise (RIN) RN from the seed laser. The approximate expression for the fraction of HOM power content ξout as a function of extracted average output power Psignal is given by [9

9. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21(2), 1944–1971 (2013). [CrossRef] [PubMed]

]
ξout=ξin(PseedPsignal)1ΓHOMΓFM(1+14RN2πΓFM|χ(Ωp)|Psignalexp[χ(Ωp)ΓFM(PsignalPseed)])
(10)
where Pseed is the seed power and Ωp is Ω for maximum χ, and ξin is the fraction of power that is initially coupled to the HOM. A seed laser is characterized by its RIN, which can be on the order of −120 dBc/Hz to −100 dBc/Hz corresponding to a laser with low and high RIN.

The HOM content vs. output power increases abruptly at the TMI threshold and 10% HOM content is used in the calculations to define the TMI threshold. Figure 6
Fig. 6 Calculated TMI threshold as a function of thermal load using two values of RIN −120 dBc/Hz (solid) and −100 dBc/Hz (dashed) representing a seed laser with low and high RIN. The crossing of signal power vs. thermal load indicates the estimated TMI threshold for the DMF85 rod fiber.
shows the calculated TMI threshold as a function of thermal load using Eq. (10). The solid and dashed curves are calculated for RN = −120 dBc/Hz and RN = −100 dBc/Hz, and represent a TMI threshold interval depending on RIN. The DMF85 rod fiber is initially SM at the signal wavelength with the SM window slightly blueshifting as the fiber heats up, and the HOM becomes guided at high thermal loads corresponding to high extracted output power. The thermal load is proportional to signal power, therefore the estimated TMI threshold occurs when the calculated TMI threshold matches the signal power used in the thermal load calculations. This happens at 371 W and 443 W, as indicated in Fig. 6. For comparison the TMI threshold of the PCF and theoretical SIF in Fig. 3(a) and Fig. 3(b) are estimated to 348 W – 426 W and 392 W – 470 W for RN = −100 dBc/Hz – RN = −120 dBc/Hz respectively. This is slightly higher for the PCF and slightly lower for the SIF compared to the DMF85, as expected based on the differential mode overlap in Fig. 3. The SM window of the DMF85 can be shifted by changing the designs parameters, which allow engineering the DMF85 rod fiber to have SM operation at high or low signal power depending on the application [15

15. M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express 20(7), 7263–7273 (2012). [CrossRef] [PubMed]

].

The TMI threshold for the DMF rod fiber has been measured to 292 W [16

16. M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express 20(5), 5742–5753 (2012). [CrossRef] [PubMed]

], which is lower than the estimated threshold. However it is comparable and the deviation is assumed tobe model dependent and not fiber design dependent, which allows the use of this model as a manufacturing tool for comparing various fiber designs.

Several system dependent parameters can be speculated to influence the estimated TMI threshold. Recent studies have also showed large dependency between TMI threshold and system parameters such as seed or pump noise as well as cooling conditions [6

6. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012). [CrossRef] [PubMed]

,19

19. A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20(22), 24545–24558 (2012). [CrossRef] [PubMed]

]. Seed noise can contain discrete frequency components possibly larger than the RIN, that if located within the frequency band of the nonlinear coupling constant reduces the TMI threshold. Any system induced loss such as photodarkening would increase the heat load and lower the TMI threshold as a function of operation time. This causes an initially stable output to become unstable as PD losses increase over time. The ability to support the HOM is highly dependent on small core refractive index changes, which could also explain a part of the deviation between calculations and measurement. It is also assumed in the calculations that the density, heat capacity, thermal conductivity, and thermo-optic coefficient are temperature independent. In experiments, the absolute temperature of the fiber increases significantly at high output power [6

6. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012). [CrossRef] [PubMed]

] and depends on outer cooling properties, which can lead to variations in these coefficients.

4. Conclusion

We have expanded a semi-analytic model to include calculations on complex microstrutered fibers with thermally induced waveguide perturbations to estimate the TMI threshold for ytterbium doped rod fiber amplifiers. The change in refractive index due to the quantum defect heating was calculated from the thermal load and used in FEM calculations to achieve mode distributions, which were combined with a semi-analytic model to estimate the TMI threshold. The DMF85 rod fiber was considered as an example with estimated TMI threshold of 371 W – 443 W, slightly higher than a previously measured TMI threshold. We believe that the presented model can be used as a manufacturing tool for comparing rod fiber performance.

References and links

1.

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]

2.

T. T. Alkeskjold, M. Laurila, L. Scolari, and J. Broeng, “Single-mode ytterbium-doped large-mode-area photonic bandgap rod fiber amplifier,” Opt. Express 19(8), 7398–7409 (2011). [CrossRef] [PubMed]

3.

F. Jansen, F. Stutzki, H.-J. Otto, M. Baumgartl, C. Jauregui, J. Limpert, and A. Tünnermann, “The influence of index-depressions in core-pumped Yb-doped large pitch fibers,” Opt. Express 18(26), 26834–26842 (2010). [CrossRef] [PubMed]

4.

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011). [CrossRef] [PubMed]

5.

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011). [CrossRef] [PubMed]

6.

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012). [CrossRef] [PubMed]

7.

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011). [CrossRef] [PubMed]

8.

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37(12), 2382–2384 (2012). [CrossRef] [PubMed]

9.

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21(2), 1944–1971 (2013). [CrossRef] [PubMed]

10.

E. Coscelli, F. Poli, T. T. Alkeskjold, M. M. Jørgensen, L. Leick, J. Broeng, A. Cucinotta, and S. Selleri, “Thermal Effects on the Single-Mode Regime of Distributed Modal Filtering Rod Fiber,” J. Lightwave Technol. 30(22), 3494–3499 (2012). [CrossRef]

11.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” J. Quant. Electron. 37(2), 207–217 (2001). [CrossRef]

12.

M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power cw fiber lasers,” Proc. SPIE 7195, 71951U, 71951U-11 (2009). [CrossRef]

13.

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express 11(22), 2982–2990 (2003). [CrossRef] [PubMed]

14.

Comsol, “Products,” <http://www.comsol.com/products/multiphysics/> (2 January 2013).

15.

M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express 20(7), 7263–7273 (2012). [CrossRef] [PubMed]

16.

M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express 20(5), 5742–5753 (2012). [CrossRef] [PubMed]

17.

F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36(23), 4572–4574 (2011). [CrossRef] [PubMed]

18.

H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20(14), 15710–15722 (2012). [CrossRef] [PubMed]

19.

A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20(22), 24545–24558 (2012). [CrossRef] [PubMed]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(140.6810) Lasers and laser optics : Thermal effects
(060.4005) Fiber optics and optical communications : Microstructured fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: April 17, 2013
Revised Manuscript: June 13, 2013
Manuscript Accepted: June 13, 2013
Published: June 20, 2013

Citation
Mette Marie Johansen, Kristian Rymann Hansen, Marko Laurila, Thomas Tanggaard Alkeskjold, and Jesper Lægsgaard, "Estimating modal instability threshold for photonic crystal rod fiber amplifiers," Opt. Express 21, 15409-15417 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15409


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References

  1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B27(11), B63–B92 (2010). [CrossRef]
  2. T. T. Alkeskjold, M. Laurila, L. Scolari, and J. Broeng, “Single-mode ytterbium-doped large-mode-area photonic bandgap rod fiber amplifier,” Opt. Express19(8), 7398–7409 (2011). [CrossRef] [PubMed]
  3. F. Jansen, F. Stutzki, H.-J. Otto, M. Baumgartl, C. Jauregui, J. Limpert, and A. Tünnermann, “The influence of index-depressions in core-pumped Yb-doped large pitch fibers,” Opt. Express18(26), 26834–26842 (2010). [CrossRef] [PubMed]
  4. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19(11), 10180–10192 (2011). [CrossRef] [PubMed]
  5. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19(14), 13218–13224 (2011). [CrossRef] [PubMed]
  6. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express20(10), 11407–11422 (2012). [CrossRef] [PubMed]
  7. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express19(4), 3258–3271 (2011). [CrossRef] [PubMed]
  8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett.37(12), 2382–2384 (2012). [CrossRef] [PubMed]
  9. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express21(2), 1944–1971 (2013). [CrossRef] [PubMed]
  10. E. Coscelli, F. Poli, T. T. Alkeskjold, M. M. Jørgensen, L. Leick, J. Broeng, A. Cucinotta, and S. Selleri, “Thermal Effects on the Single-Mode Regime of Distributed Modal Filtering Rod Fiber,” J. Lightwave Technol.30(22), 3494–3499 (2012). [CrossRef]
  11. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” J. Quant. Electron.37(2), 207–217 (2001). [CrossRef]
  12. M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power cw fiber lasers,” Proc. SPIE7195, 71951U, 71951U-11 (2009). [CrossRef]
  13. J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express11(22), 2982–2990 (2003). [CrossRef] [PubMed]
  14. Comsol, “Products,” < http://www.comsol.com/products/multiphysics/ > (2 January 2013).
  15. M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express20(7), 7263–7273 (2012). [CrossRef] [PubMed]
  16. M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express20(5), 5742–5753 (2012). [CrossRef] [PubMed]
  17. F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett.36(23), 4572–4574 (2011). [CrossRef] [PubMed]
  18. H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express20(14), 15710–15722 (2012). [CrossRef] [PubMed]
  19. A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express20(22), 24545–24558 (2012). [CrossRef] [PubMed]

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